Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations
Abstract
:1. Introduction
- (1)
- It allows us to extend collocations and mechanical quadratures methods to hypersingular integral equations with non-Riemann integrable right sides;
- (2)
- For linear hypersingular integral equations, it allows one to verify the inverse operator existence and estimate its norm quite easily;
- (3)
- The method is stable with respect to the operator and right hand side perturbations;
- (4)
- The method does not require the existence and reversibility of the nonlinear operator derivative.
2. Continuous Method and Its Convergence Properties
Continuous Method for Solving Operator Equations
- 1.
- The inequality
- 2.
- Inequality (12) is satisfied.
3. An Solution of Hypersingular Integral Equations with the Continuous Method
- (i)
- The above limit exists;
- (ii)
- has at least p derivatives in the neighborhood of the point .
3.1. An Approximate Solution of Linear Hypersingular Integral Equations with Second Order Singularity
- (1)
- Unite the nodes and , denoting them by
- (2)
- Unite the nodes and , denoting them by
- (3)
- Denote the family of basis functions , by ;
- (4)
- Denote by unknowns , .
3.2. Nonlinear Hypersingular Integral Equations
4. Summary and Discussion
- (1)
- The method is applicable for solving linear and nonlinear hypersingular integral equations, whose right-hand sides contain non-Riemann integrable functions.
- (2)
- In Section 3.1 the continuous method is applied to linear hypersingular integral equations with the singularities of the second order. The conditions for the unique solvability of the constructed computing scheme are obtained and the convergence of the sequence of approximate solutions to the exact one is proven. It is shown that for linear hypersingular integral equations, the method converges for sufficiently large N and for .
- (3)
- In Section 3.2 the continuous method is applied to nonlinear hypersingular integral equations with the singularities of the second order. Conditions are given for the convergence of the constructed iterative spline-collocation method to the solution of a nonlinear hypersingular integral equation. It should be noted that the method is applicable to hypersingular integral equations of the first and second kinds.
- 1.
- To obtain a set of convergence conditions owing to logarithmic norm values in various spaces;
- 2.
- To determine the norm of the inverse matrix of an approximate system;
- 3.
- To determine stability boundaries for solutions with respect to variations of kernels and right-hand sides of the equations.
Author Contributions
Funding
Conflicts of Interest
References
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Boykov, I.; Roudnev, V.; Boykova, A. Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations. Axioms 2020, 9, 74. https://doi.org/10.3390/axioms9030074
Boykov I, Roudnev V, Boykova A. Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations. Axioms. 2020; 9(3):74. https://doi.org/10.3390/axioms9030074
Chicago/Turabian StyleBoykov, Ilya, Vladimir Roudnev, and Alla Boykova. 2020. "Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations" Axioms 9, no. 3: 74. https://doi.org/10.3390/axioms9030074